m_d_lawson

m_d_lawson computes the rational approximation of a matrix-valued function F.

Summary:

m_d_lawson computes the rational approximation of a matrix-valued function F. In the implementation, F is vectorized by

F = [f1, ..., fk]: C--> C^k.

1). The underlying problem that m_d_lawson solves is

min_{P,q} max_j ||F(x_j)-P(x_j)./q(x_j)||_{fro}^2

1<=j<=m

which can also be written as

min_{p1,...,pk,q} max_j (|f1-p1/q|^2 + ... + |fk - pk/q|^2)_j.

1<=j<=m

2). m_d_lawson is based on solving the dual problem

max_{w: w^Te = 1, w \ge 0} d(w)

where d(w) is the dual function and w is updated by Lawson's method.

3). Function:

[Rfun,relgap,dk,err,pol,zers] = m_d_lawson(x1, bigF, nd, deg_vec, maxit);

% Input:

% x1 m sample points;

% bigF sampled data matrix of F or its vectorized;

% nd the degree of denominator polynomial;

% deg_vec degrees (n1,n2,...,nk) of numerators P=[p1,...,pk

% maxit maximum number of Lawson's iterations

%

% Output:

% Rfun cellarray with Rfun{1} = r1,...,Rfun{ell} = r_ell

% each r_k = p_k/q is a function handle of type (nk,nd)

% pol zeros of q;

% zers cellarray with zers{k} is zeros of pk;

% dk the sequence of the dual function values d(w) at w;

% err the record of maximun F-norm errors, i.e.,

% err(k) = MAX_j ||bigF(j,:)-bigR(j,:)||_{fro};

% relgap relative daulity gap at the computed approximant;

% generally, relgap < 0.01 means the computed can be

% viewed as the minimax approximant;

% Remark:

% This code is also suitable for the scalar-value function, as well as

% the degree-specified duplexer problem.

4). Examples (refer to):

test_mdlawson_1_bestrat.m

test_mdlawson_2_bestpoly.m

test_mdlawson_3_converg.m

5). References:

[1] L.-H. Zhang, Y.-N. Zhang, C. Zhang and S. Han, Rational minimax approximation of matrix-valued functions, 2025.

[2] L.-H. Zhang, L. Yang, W. H. Yang and Y.-N. Zhang, A convex dual problem for the rational minimax approximation and Lawson's iteration, Math. Comp. 94 (2025), 2457?2494.

[3] L.-H. Zhang and S. Han, A convergence analysis of Lawson's iteration for computing polynomial and rational minimax approximations, 2025, SIAM J. Numer. Anal., https: //arxiv.org/pdf/2401.00778v3, To appear.